We consider the free boundary problem for a 2D and 3D fluid filtered in porous media, which is known as the one-phase Muskat problem. We show that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a unique global Lipschitz strong solution. The proof of the uniqueness relies on a new pointwise $C^{1,\alpha}$ estimate near the boundary for harmonic functions. This is based on joint work with Francisco Gancedo (Universidad de Sevilla) and Huy Q. Nguyen (University of Maryland).

嘉宾简介：Hongjie Dong is a Professor at the Division of Applied Mathematics of Brown University. He received his Ph.D. in 2005 at the Department of Mathematics in the University of Minnesota. Prof. Dong's current research interests include Partial Differential Equations, Probability, and Numerical Analysis. He was the recipient of an NSF early career award in 2011 and a Simons fellowship in 2021. He has also been on the editorial boards of several journals including SIAM J. Math. Anal. and J. Differential Equations.